We just got home from Europe but not the UK. We returned from Amsterdam and had the same concerns as you mentioned. We also needed to be tested for my return to the USA. We were also concerned about finding testing sites before our departure and I found some in Amsterdam for our use. Antigen tests are fine. However, they have to be done less than 24 hrs before departure.
I did some research for you and the UK and read your posts and think the following information may help you.
First, here is a link for requirements to the USA for Covid tests. Basically the Antigen is the best option but only with a 24 hr option. https://www.msn.com/en-gb/news/newsbirmingham/usa-travel-rules-what-covid-test-do-i-need-to-travel-to-the-usa/ar-AAWnjNN?ocid=uxbndlbing
Second, if I were in the UK I would schedule my test appointment in advance (you usually have to prepay for the test too). I looked up this site for testing and here are some that may help. If you can do the test before you flight, I would choose one to the sites at Heathrow. https://www.expresstest.co.uk/book-a-test/?msclkid=f204c7d434d41fb36a4ba5b450c8744c&utm_source=bing&utm_medium=cpc&utm_campaign=Outbound Other - UK - Generic - Exact&utm_term=covid test heathrow&utm_content=UK - Generic | Heathrow
I also clicked on the first link for the 35 Euro test (the one I think you want) and here were the results for the pulldown menu with the sites available. If you scroll down you will see some at Heathrow. However, other sites are listed and they may be good too. There are also directions for each site, so be sure you can find the site and it is convenient before you book.
If you cannot get a document in person, then you should get an email or text to your phone with the results. Within either the email or test you should either get a formal document with a QR code, or a link to get this for use at the airport. If you can print this that would be best, if not, most airlines accept the cell phone document with the QR, but please check with your airline first.
You will also need to have proof of Covid vaccination. I assume you already have that.
I hope this helps.